Finite Group Databasebeta

← #26111334

Subgroups — #26111334

Order 12032.

660 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689054C211yesmaximal, characteristic
2#20246005C211yesmaximal, characteristic
3#20243474C211yesmaximal, characteristic
4C2×#15688848C211yesmaximal, characteristic
5#20241268C211yesmaximal, characteristic
6#20241266C211yesmaximal, characteristic
7C47×#1823C211yesmaximal, characteristic
8C2×#13093980—21no—
9C22×Dic47⋊C4—21no—
10C22×Dic47⋊C4C2211yescharacteristic
11C2×C22⋊(C47⋊C8)C2211yescharacteristic
12#15689054—22no—
13#15689054C414yescomplemented
14#15689020—22no—
15#15689018—22no—
16#15688983C2211yescharacteristic
17#15688981—21no—
18C2×C47⋊#95—21no—
19C2×C47⋊#95C2211yescharacteristic
20#15688885—22no—
21#15688848—22no—
22C22×C4⋊C4×C47C2211yescharacteristic
23C2×C22⋊C8×C47C2211yescharacteristic
24C47×#380C2211yescharacteristic
25C47×#340—22no—
26C23×Dic47—21no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C22×C47⋊C8—21no—
30#13093980—42no—
31#13093980—42no—
32C2×C4⋊Dic47D412yes—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—22no—
38C2×Dic47⋊C4—24no—
39C2×Dic47⋊C4—22no—
40C2×Dic47⋊C4—42no—
41C2×Dic47⋊C4—42no—
42C2×Dic47⋊C4C2×C412yes—
43C2×Dic47⋊C4D412yes—
44C2×C22⋊Dic47—21no—
45C2×C4×Dic47—22no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C47⋊#95—22no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C47⋊#95—24no—
52C47⋊#95C2×C412yes—
53C23×C4×C47—21no—
54C23×C4×C47C2311yescharacteristic
55C22×C8×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47C2×C412yes—
62C2×C4⋊C4×C47D412yes—
63C2×C22⋊C4×C47—21no—
64C22⋊C8×C47—21no—
65C22⋊C8×C47—21no—
66#95×C47—21no—
67C24×C47C2×D411yescharacteristic
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—42no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4111yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—24no—
88C22×Dic47—22no—
89C22×Dic47—21no—
90C22×Dic47—21no—
91C22×Dic47—22no—
92C22×Dic47—42no—
93C22×Dic47—41no—
94C22×Dic47—41no—
95C22×Dic47—42no—
96C22×Dic47C22⋊C412yes—
97C22×Dic47C22⋊C412yes—
98C22×Dic47D812yes—
99C22×Dic47SD1612yes—
100C2×C8×C47—42no—
101C2×C8×C47—41no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C4⋊C4×C47—42no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—41no—
109C22⋊Dic47—41no—
110C22⋊Dic47—41no—
111C4⋊Dic47—42no—
112Dic47⋊C4—22no—
113Dic47⋊C4—22no—
114Dic47⋊C4—22no—
115Dic47⋊C4—22no—
116Dic47⋊C4—44no—
117Dic47⋊C4—48no—
118Dic47⋊C4—82no—
119C4×Dic47—44no—
120C2×C47⋊C8—42no—
121C2×C47⋊C8—41no—
122C23×C47—22no—
123C23×C47—22no—
124C23×C47C2×C22⋊C411yescharacteristic
125C23×C47#12011yescharacteristic
126C23×C47#12311yescharacteristic
127C23×C47C2×D811yescharacteristic
128C23×C47C2×SD1611yescharacteristic
129C23×C47AGL(1,ℤ/8)11yescharacteristic
130C23×C47#13711yescharacteristic
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—81no—
144C2×C4×C47—81no—
145C2×C4×C47—81no—
146C2×C4×C47—81no—
147C2×C4×C47C4×D411yescharacteristic
148C2×C4×C47C4×D411yescharacteristic
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47#12111yescharacteristic
151C2×Dic47—22no—
152C2×Dic47—22no—
153C2×Dic47—22no—
154C2×Dic47—22no—
155C2×Dic47—22no—
156C2×Dic47—22no—
157C2×Dic47—24no—
158C2×Dic47—22no—
159C2×Dic47—44no—
160C2×Dic47—44no—
161C2×Dic47—42no—
162C2×Dic47—42no—
163C2×Dic47—42no—
164C2×Dic47—42no—
165C2×Dic47—42no—
166C2×Dic47—42no—
167C2×Dic47—42no—
168C2×Dic47—81no—
169C2×Dic47#10214yes—
170C8×C47—81no—
171C47⋊C8—81no—
172#9027—471nomaximal, Sylow
173C22×C47—22no—
174C22×C47—22no—
175C22×C47—22no—
176C22×C47—22no—
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47#38611yescharacteristic
182C22×C47C2×#10211yescharacteristic
183C22×C47#41711yescharacteristic
184C22×C47#44711yescharacteristic
185C22×C47#44811yescharacteristic
186C22×C47#45612yes—
187C22×C47#45812yes—
188C22×C47#48011yescharacteristic
189C22×C47#48111yescharacteristic
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—21no—
194C4×C47—82no—
195C4×C47—82no—
196Dic47—24no—
197Dic47—44no—
198Dic47—42no—
199Dic47—82no—
200C2×#398—471no—
201#2316—471no—
202#1823—471no—
203#1823D4711yescharacteristic
204C2×#324—471no—
205#1294—471no—
206#1294—471no—
207C2×C47—21no—
208C2×C47—21no—
209C2×C47—21no—
210C2×C47—21no—
211C2×C47#161012yes—
212C2×C47#161112yes—
213C2×C47#187411yescharacteristic
214C2×C47#188611yescharacteristic
215C2×C47#189011yescharacteristic
216C2×#117—941no—
217C22×C4⋊C4—471no—
218C22×C4⋊C4—941no—
219C22×C4⋊C4C2×D4711yescharacteristic
220C2×C22⋊C8—471no—
221C2×C22⋊C8C2×D4711yescharacteristic
222#398—474no—
223#398—942no—
224#389—942no—
225#389—942no—
226#380—471no—
227#380—941no—
228#380C2×D4711yescharacteristic
229C2×#95—471no—
230C2×#95—941no—
231#340—22no—
232#340—942no—
233#324—942no—
234C47#902711yescharacteristic, complemented, Sylow
235C23×C4—21no—
236C23×C4—471no—
237C23×C4—471no—
238C23×C4—941no—
239C23×C4C22×D4711yescharacteristic
240C22×C8—21no—
241C22×C8—941no—
242#117—1882no—
243#117—1882no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—472no—
250C2×C4⋊C4—472no—
251C2×C4⋊C4—472no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4—941no—
255C2×C4⋊C4—941no—
256C2×C4⋊C4—942no—
257C2×C4⋊C4—944no—
258C2×C4⋊C4—942no—
259C2×C4⋊C4—1882no—
260C2×C4⋊C4—1882no—
261C2×C4⋊C4C4×D4712yes—
262C2×C4⋊C4C22⋊D4712yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—941no—
265C2×C42—942no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—941no—
269C22⋊C8—941no—
270#95—21no—
271#95—472no—
272#95—944no—
273#95—942no—
274#95—941no—
275#95—941no—
276C24D4×D4711yescharacteristic
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—472no—
287C22×C4—472no—
288C22×C4—472no—
289C22×C4—472no—
290C22×C4—941no—
291C22×C4—941no—
292C22×C4—941no—
293C22×C4—941no—
294C22×C4—941no—
295C22×C4—944no—
296C22×C4—942no—
297C22×C4—941no—
298C22×C4—942no—
299C22×C4—1882no—
300C22×C4—1882no—
301C22×C4—1881no—
302C22×C4—1881no—
303C22×C4C2×C4×D4711yescharacteristic
304C22×C4#1062535011yescharacteristic
305C22×C4D4×D4711yescharacteristic
306C22×C4D4×D4711yescharacteristic
307C22×C4#1062535211yescharacteristic
308C22×C4C2×C22⋊D4711yescharacteristic
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—1882no—
312C2×C8—1881no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—942no—
319C4⋊C4—942no—
320C4⋊C4—942no—
321C4⋊C4—942no—
322C4⋊C4—3762no—
323C4⋊C4—1882no—
324C4⋊C4—1888no—
325C4⋊C4—1884no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—1881no—
329C22⋊C4—1881no—
330C42—1884no—
331C23—22no—
332C23—22no—
333C23C22⋊C4×D4711yescharacteristic
334C23#1309399211yescharacteristic
335C23#1309400611yescharacteristic
336C23D8×D4711yescharacteristic
337C23SD16×D4711yescharacteristic
338C23#1309404811yescharacteristic
339C23#1309405311yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—474no—
357C2×C4—942no—
358C2×C4—942no—
359C2×C4—942no—
360C2×C4—942no—
361C2×C4—942no—
362C2×C4—942no—
363C2×C4—944no—
364C2×C4—942no—
365C2×C4—3761no—
366C2×C4—1882no—
367C2×C4—1882no—
368C2×C4—1882no—
369C2×C4—1882no—
370C2×C4—1884no—
371C2×C4—1884no—
372C2×C4—1882no—
373C2×C4—1882no—
374C2×C4—1882no—
375C2×C4(C22⋊D47)⋊C411yescharacteristic
376C2×C4C4×C22⋊D4711yescharacteristic
377C2×C4#1309399511yescharacteristic
378C2×C4#1309399711yescharacteristic
379C8—81no—
380C8—3761no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22#1568899911yescharacteristic
390C22#102×D4711yescharacteristic
391C22#1568912111yescharacteristic
392C22#1568925511yescharacteristic
393C22#1568926011yescharacteristic
394C22#1568930412yes—
395C22#1568931212yes—
396C22#1568943511yescharacteristic
397C22#1568944011yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—82no—
403C4—82no—
404C4—944no—
405C4—3762no—
406C4—1884no—
407C4—1882no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#2024238912yes—
413C2#2024239412yes—
414C2#2024370211yescharacteristic
415C2#2024377811yescharacteristic
416C2#2024379811yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1823C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1823D47
hypocenterC47#9027
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
13. #15689054C4semidirect
234. C47#9027semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156890542
#202460052
#202434742
C2×#156888482
#202412682
#202412662
C47×#18232
#902747